This article investigates the performance of grid computing systems whose interconnections are given by random and scale-free complex network models. Regular networks, which are common in parallel computing architectures, are also used as a standard for comparison. The processing load is assigned to the processing nodes on demand, and the efficiency of the overall computing is quantified in terms of the respective speed-ups. It is found that random networks allow higher computing efficiency than their scale-free counterparts as a consequence of the smaller number of isolated clusters implied by the former model. At the same time, for fixed cluster sizes, the scale free model tend to provide slightly better efficiency. Two modifications of the random and scale free paradigms, where new connections tend to favor more recently added nodes, are proposed and shown to be more effective for grid computing than the standard models. A well-defined correlation is observed between the topological properties of the network and their respective computing efficiency.